Physical systems can be modeled mathematically to simulate their behavior under certain conditions. There are a wide variety of means to model these systems, ranging from the very simplistic to the extremely complicated. One of the more sophisticated means to model physical systems is through the use of finite element analysis. As the name implies, finite element analysis involves the representation of individual, finite elements of the physical system in a mathematical model and the solution of this model in the presence of a predetermined set of boundary conditions. Another comparable means to model physical systems is through the use of finite difference analysis. Finite difference analysis involves the modeling of individual points within a modeled space and computing the differences between these points. Finite difference analysis is often used for simulating the dynamic behavior of fluids.
Traditional finite difference techniques and finite element techniques using streamlined, upwinding methods are typically used to simulate the production of oil in a reservoir. While each of these techniques has its own advantages, it also has its own disadvantages. Generally speaking, the finite difference techniques produce physically realistic results, but they are not very accurate. The finite element techniques, on the other hand, are more accurate, but they produce results which are not physically realistic.
As a result of their respective disadvantages, both the finite difference techniques and finite element techniques which are conventionally used normally require a great deal of computer resources. In the case of finite difference techniques, reasonable accuracy can be achieved, but this requires many more nodes then would be necessary in a finite element model. This increases the amount of memory and CPU time which are needed to compute an accurate solution. In the case of finite element techniques, comparable accuracy can be achieved with fewer nodes, but the solutions may not be realistic. For example, concentrations may be more than 100 percent, or permeabilities may be negative. Consequently, there may be problems for which the solution does not converge, or for which the solution may converge very slowly. These techniques are therefore less reliable and may require a large number of iterations before acceptable accuracy is achieved.
Because each of these standard techniques has its own drawbacks, and because these drawbacks increase in the amount of computer resources which are necessary to generate acceptable solutions, it would be desirable to provide a method for modeling systems such as oil reservoirs which reliably produces accurate, realistic solutions for these systems.